A 100 e Problem
Notation:

For A ∈ IRn×n , x ∈ IRn denote by |A|, |x| the matrix, vector of absolute values.
nHence |A| ∈ IRn×n≥0 , |x| ∈ IR≥0 .

For x, y ∈ IRn comparison is entrywise, i.e., x ≥ y :⇔ xi ≥ yi for i ∈ {1, ..., n}.
Denote e ∈ IRn with e := (1, ..., 1)T .
Conjecture:A ∈ IRn×n with |A|e = ne ⇒ ∃ 0 6= x ∈ IRn : |Ax| ≥ |x|.

The geometrical interpretation of the assumptions is that the rows of A lie on the octahedron
centered at the origin with the nonzero entry of the vertices equal to n.
For a proof or counterexample of the conjecture I am happy to reward you with 100 e.
Note:

Theorem 5.8 in [8] proved
A ∈ IRn×n with |A|e = ne ⇒ ∃ 0 6= x ∈ IRn : |Ax| ≥ 3+21√2 |x|.

A stronger conjecture
Denote the i-th row of A by ri . Dr. Florian Bünger proposed the following stronger formulation:
Conjecture 2:

A ∈ IRn×n , n ≥ 2 and kri k2 ≥√n − 1 for all i ∈ {1, . . . , n} ⇒n∃ 0 6= x ∈ IR : |Ax| ≥ |x|.
√This formulation is stronger because |A|e = ne ⇒ n = |ri |e ≤ kri k2 n.
The geometrical interpretation of the assumptions of Conjecture 2 is that the rows of A lie√
on or outside the sphere centered at the origin with radius n − 1.
This second conjecture is particularly appealing because if true it is sharp in the following sense:
For any α > 1 Conjecture 2 does not hold true when replacing the assertion by |Ax| ≥ α|x|.
A counterexample is the n × n matrix A with zero diagonal, all elements above thediagonal equal to 1 and all elements below the diagonal equal to −1.
The matrix satisfies the assumptions, and |Ax| = |x| for x = (1, 1, 0, . . .)T .
However, it is shown in [8, Lemma 5.7] that there is no x ∈ IRn with |Ax| > |x|.
This sharpness of Conjecture 2 may offer more efficient proof schemes.
I am happy to reward you with 100 e for a proof or counterexample of Conjecture 2.
That implies that a proof of Conjecture 2 is worth 200 e.

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Theoretical background
For IK ∈ {IR≥0 , IR, C} and a matrix A ∈ IKn×n consider the quantity(0.1)%IK (A) := max{|λ| : λ ∈ IK, 0 6= x ∈ IKn }.

For IK = IR≥0 Perron-Frobenius Theory implies that this is the Perron root of a nonnegative matrix. ForIK = IR this quantity was introduced in [8] as the sign-real spectral radius, and for IK = C it was introducedin [11] and called the sign-complex spectral radius.
The sign-real spectral radius originated in the investigation of the componentwise distance to the nearest
singular matrix [9, 7]. One reason why %IK gained interest in matrix theory was that Collatz’s [4] characterization of the Perron root extends to the sign-real and the sign-complex spectral radius:(0.2)

A ∈ IKn×n :%IK (A) = maxn minx∈IK

x6=0xi 6=0(Ax)i.
xi

Moreover, all three quantities %IK share a number of other quantities with the Perron root:%IK (AH ) = %IK (A)|S1 | = |S2 | = I⇒ %IK (S1 AS2 ) = %IK (A)P ∈ IRn×n permutation matrix⇒ %IK (P T AP ) = %IK (A)D ∈ IKn×n nonsingular diagonal⇒ %IK (D−1 AD) = %IK (A)α ∈ IK ⇒ %IK (αA) = |α|%IK (A)µ ⊆ {1, . . . , n}

⇒ %IK (A[µ]) ≤ %IK (A) [inheritance]

A triangular ⇒ %IK (A) = maxi |Aii |The %IK are continuous [8], and in [13] it was shown that the first four linear transformations and their
combinations capture all linear operators leaving %IR invariant. A number of other properties of %IK are
discussed in [3, 5, 6, 12] and the literature cited over there. Bounds using cycle products were investigated[10, 1], and in [2] a number of topological, invariance and other properties are discussed.
REFERENCES[1] F. Bünger. A short note on the ratio between sign-real and sign-complex matrices of a real square matrix. Linear Algebra
and its Applications (LAA), 529:126–132, 2017.
[2] F. Bünger and A. Seeger. On sign-real spectral radii and sign-real expansive matrices. Linear Algebra and its Applications(LAA), 680:293–324, 2024.
[3] A. Goldberger and M. Neumann. Perron-Frobenius theory of seminorms: a topological approach. Linear Algebra and itsApplications (LAA), 399:245–284, 2005.
[4] L. Collatz. Einschließungssatz für die charakteristischen Zahlen von Matrizen. Math. Z., 48:221–226, 1942.
[5] J.M. Peña. A characterization of the distance of infeasibility under block-structered perturbations. Linear Algebra andits Applications (LAA), 370:193–216, 2003.
[6] M. Radons. Direct solution of piecewise linear systems. Theor. Comp. Sc., 626:97–109, 2016.
[7] S.M. Rump. Almost Sharp Bounds for the Componentwise Distance to the Nearest Singular Matrix. Linear and Multilinear
Algebra (LAMA), 42:93–107, 1997.
[8] S.M. Rump. Theorems of Perron-Frobenius type for matrices without sign restrictions. Linear Algebra and its Applications(LAA), 266:1–42, 1997.
[9] S.M. Rump. Bounds for the Componentwise Distance to the Nearest Singular Matrix. SIAM J. Matrix Anal. Appl.
(SIMAX), 18(1):83–103, 1997.
[10] S.M. Rump. The sign-real spectral radius and cycle products. Linear Algebra and its Applications (LAA), 279:177–180,
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[11] S.M. Rump. Perron-Frobenius Theory for Complex Matrices. Linear Algebra and its Applications (LAA), 363:251–273,
2003.
[12] X. Shi and Y. Wei. A sharp version of Bauer-Fike’s theorem. J. Comput. Appl. Math., 236(13):3218–3227, 2012.
[13] B. Zalar. Linear operators preserving the sign-real spectral radius. Linear Algebra and its Applications (LAA), 301(13):99–112, 1999.
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